T-CONTINUOUS-INFORMATION: Regularization, Stopping and Predictable Finite Variation

Work on a finite interval [0,K] with a complete right-continuous filtration. Completeness means that F₀ contains all P-null sets of the ambient σ-algebra and their subsets. We use only discrete martingales, conditional expectation, pathwise finite-variation integration, and the established integral limit theorems.

1. First Construct Paths That Can Be Read at Random Times

For Y∈L¹, Mₜ=E[Y|Fₜ] has a càdlàg modification. More generally, a class D submartingale S has a càdlàg modification if t↦ESₜ is right-continuous. On this interval class D means that Sτ, over all stopping times τ≤K, form a uniformly integrable family. The broader version is used only for projections of raw increasing processes in this chapter.

Take nested finite grids containing 0 and K whose union D is dense. Apply the Chapter 15 upcrossing inequality on each grid. The endpoint L¹ bound gives a common finite bound on the expected number of upcrossings of every rational level interval. Monotone convergence excludes infinitely many such upcrossings on D. Escape to infinity must also be excluded: stop at the first grid point with |S|>L, or at K if there is none. The uniform L¹ bound supplied by class D makes the hit probability at most C/L. Increase the grids and then send L to infinity. Outside one null set, the path on D is therefore bounded and has finite one-sided limits everywhere; otherwise two interlaced approaching sequences would force infinitely many upcrossings of a rational level interval.

Define S⁺ₜ by the right limit for t<K and keep the given S_K. The deterministic criterion of finite one-sided limits on a dense set gives a càdlàg extension: a failure of right continuity would let us select further dense points contradicting the defining right limit. S⁺ₜ is Fₜ₊-measurable, hence adapted by right continuity of the filtration. Class D gives L¹ convergence along decreasing deterministic times. The submartingale relation yields Sₜ≤E[S⁺ₜ|Fₜ]=S⁺ₜ; right continuity of the means makes their expectations equal, so S⁺ₜ=Sₜ almost surely. Completeness lets us set entire paths to zero on the common exceptional set while retaining adaptedness.

For a closed martingale M, the required class D bound must be verified without circularly using paths not yet constructed. First use grid stopping times only: finite splitting gives Mτ=E[Y|Fτ]. Conditional expectations of one Y over arbitrary sub-σ-algebras form a uniformly integrable family: their tail events have uniformly small probabilities, and the integral of |Y| is uniformly absolutely continuous over small-probability sets. The preceding path argument therefore applies to M. For an arbitrary τ, now read the càdlàg path at decreasing grid times τₙ. Uniform integrability and Vitali give L¹ convergence and yield the identity

Explicitly, for B∈Fτ⊂Fτₙ, the grid identity E[\mathbf1_B M_{\tau_n}]=E[\mathbf1_B Y] passes to the L¹ limit. This also proves uniform integrability of the entire bounded stopped family of a closed martingale. For a càdlàg class D submartingale, bounded optional sampling follows in the same way from grid inequalities, uniform integrability and right continuity, without stochastic integration.

2. Generators of the Predictable σ-Algebra and Reading at Stops

The predictable σ-algebra P is generated by left-continuous adapted real processes, equivalently by F₀×{0} and B×(s,t], where B∈Fₛ. In one direction their indicator processes are left-continuous and adapted. In the other, replace a left-continuous process on each grid interval (s,t] by its left-endpoint value and pass to the pointwise limit.

For any stopping time T, define \mathcal F_{T-}=σ(F₀, B∩{s<T}:B∈Fₛ,s≥0). On {T<∞}, H_T is \mathcal F_{T-}-measurable whenever H is predictable: check the rectangular generators first, then use the functional monotone class theorem. Handle H₀ separately through F₀. T itself need not be announceable.

The optional σ-algebra O is generated by càdlàg adapted real processes, and P⊂O: approximate left-continuous processes by right-continuous step processes using left grid values. A càdlàg adapted process is progressively measurable: on each [0,t], approximate by right grid values not exceeding t to obtain B([0,t])⊗Fₜ measurability. Optional processes consequently have \mathcal F_T-measurable values at stopping times. For the last step compose the measurable function on [0,t] with ω↦(T∧t,ω) and restrict to {T≤t}; no general début theorem for random sets is needed.

3. Why a Predictable Finite-Variation Martingale Is Constant

Let D be a càdlàg predictable martingale, with D₀=0 and integrable pathwise total variation V_K. Integrate pathwise in the Stieltjes sense. For rectangular simple predictable H, E∫H dD=0 because each term is a bounded earlier-measurable variable times a future martingale increment. Integrability of total variation extends the identity, by the monotone class theorem for finite signed measures, to every bounded predictable H. Measurability of path integrals is first checked on rectangles and then extended using the positive and negative variation measures.

Take f(x)=√(1+x²), a convex function with bounded derivative. Sum its supporting-tangent inequality on any deterministic partition and refine to obtain the inequality below. By right continuity of D, right-endpoint step values converge pointwise to f′(Dₜ); dominated convergence with respect to the path variation measure applies. This is ordinary finite-variation integration, not Itô formula.

Since f′(D) is predictable, the expectation identity really does apply. The left side is nonnegative and vanishes only when D_K=0, so D_K=0 almost surely. Repeat at rational terminal times and use right continuity to obtain indistinguishability from zero. In particular, two predictable increasing processes starting at zero with integrable terminal values are indistinguishable if their difference is a martingale. This is the full uniqueness argument used in Chapter 30.

Predictability cannot be dropped: the compensated Poisson martingale Nₜ−λt has integrable total variation on finite intervals and is not constant. The result must not be misread as saying that every finite-variation local martingale is constant.

4. At an Announceable Time, a Martingale Jump Has No Prior Bias

Let 0<T≤K have announcing stopping times Sₙ<T with Sₙ↑T, and let M be a closed martingale. Càdlàg paths and uniform integrability give M_{S_n}\to M_{T-} in L¹. For B=A∩{s<T}, A∈Fₛ, use Bₙ=A∩{s<Sₙ}∈\mathcal F_{S_n}. Bounded sampling gives E[\mathbf1_{B_n}(M_T-M_{S_n})]=0. Pass to the limit and apply the monotone class theorem to the generators to obtain

For F₀ generators take Bₙ=B directly. If S=M+A is the Chapter 30 decomposition, both A_T and A_{T-} are \mathcal F_{T-}-measurable, and hence \Delta A_T=E[\Delta S_T\mid\mathcal F_{T-}]. Predictable compensation is therefore not the same as continuous compensation: an anticipated time may still carry a jump with a conditional mean.

This page does not use, and does not claim to prove, general measurable projection, or début and section theorems for arbitrary random sets. Return to Chapter 30.